water logged lawn

hi back again ive recently started renting a house with a good size garden all down 1 side and along the back are some big conefer trees and a couple of other types ones thrown in, but ive now noticed alot of water gathering on the lawn the lawn is clayey, what can i do to try and help the lawn theres alot of moss growing on the lawn. the other problem is the soil is full of roots and dry around the edges under the trees what could i plant in this type of soil or should i be digging up the rootsand turning the soil.
thanks for the advice, andrew